Exercise 4
Linear Elasticity Weak Form
We consider the momentum balance in static equilibrium
\[ \nabla \cdot \mathbf{\sigma} = \mathbf{f} \]
where \(\mathbf{\sigma}\) is the Cauchy stress tensor and \(\mathbf{f}\) are external forces applied to the system. This equation is the basis for many structural engineering problems.
It is the goal of this exercise to derive a weak formulation of the problem in a simplified setting. We assume Hooke’s law in it’s simplest form and linearly relate the stress \(\mathbf{\sigma}\) to the strain \(\mathbf{\epsilon}= \nabla \mathbf{w}\), where \(\mathbf{\epsilon}\) is the strain, \(\mathbf{w}\) is the displacement and \(c \in \mathbb{R}\) being the stiffness.
\[ \mathbf{\sigma} = c \; \mathbf{\epsilon} = c \; \nabla \mathbf{w} \]
Given the rectangular domain \(\Omega\) shown in the figure with Dirichlet boundaries denoted with \(\Gamma_D\) and Neumann boundaries denoted with \(\Gamma_N\), we state the following strong formulation of our linear elastics problem
\[ \begin{cases} \begin{aligned} & c \nabla \cdot \nabla \mathbf{w} = \mathbf{f} \\ & \mathbf{w} = \mathbf{g} \\ & \nabla \mathbf{w} \cdot \mathbf{n} = \mathbf{h} \\ \end{aligned} \quad \begin{aligned} & \quad \text{on} \; \Omega \\ & \quad \text{on} \; \Gamma_D \\ & \quad \text{on} \; \Gamma_N \end{aligned} \end{cases} \]
Be precise about the dimensions of each object in your derivations. Note that \(\mathbf{w} \in \mathbb{R}^2\) for this two dimensional example.
Tasks
Derive the weak formulation of the problem assuming that the displacement \(\mathbf{w} \in H^1(\Omega)\) and test function \(\mathbf{v} \in H^1(\Omega)\). Make sure to differentiate between the Dirichlet and Neumann boundary term. Why is the implementation of the Dirichlet term in this weak formulation difficult?
Recall the very common notation that a function space with subscript zero, e.g. \(\mathbb{V}_0\) denotes compact support. For our purposes, this means that the function vanishes at the boundary.
Recall from your PDE class that \(H^1\) denotes a Hilbert space.
Recall from your PDE-class that a weak formulation is derived by multiplying your problem with a test function (e.g. denoted as \(\mathbf{v}\)) and integrated over the domain \(\Omega\). Often, integration by parts is employed to reduce the demands on the funtional space of the solution.
Your weak formulation always needs to be stated in a way similar to the following:
Find \(\mathbf{w} \in \mathbb{V}\) such that \[ \begin{cases} & \text{YOUR } \\ & \text{PROBLEM} \\ & \text{STATEMENT} \\ \end{cases} \] \(\forall \mathbf{v} \in \mathbb{V}_0\)
Where \(\mathbf{v} \in \mathbb{V}_0\) denotes the test function we used to derive the weak form.
Isotropy and Objectivity
The Cauchy Stress tensor \(\mathbf{\sigma}\) given in the deformed configuration can be transformed in the reference (undeformed) configuration with the Second-Piola Kirchoff Stress tensor \(\mathbf{S}\). The following relation holds
\[ \mathbf{S} = \text{det}(\mathbf{F}) \mathbf{F}^{-1} \mathbf{\sigma} \mathbf{F}^{-T} \] and is called the pull-back operation.
Consider the following material models or constitutive relations:
- \(\mathbf{S} = \alpha(\mathbf{C} - \mathbf{I})\)
- \(\mathbf{\sigma} = \alpha \mathbf{F}\)
- \(\mathbf{S} = \alpha \mathbf{I} + 2\beta [\text{tr}(\mathbf{B}) - 3]\mathbf{I}\)
where, \(\mathbf{C}\) and \(\mathbf{B}\) are the Right and Left Cauchy-Green tensors, \(\alpha, \beta\) are scalar parameters, and \(\mathbf{I}\) is the identity tensor.
Tasks
- Decide whether each of the material model is objective, and give reasons for the decision.
- Decide whether each of the material model is isotropic, and give reasons for the decision.
Polar Decomposition of Deformation Gradient
Let
\[ B = \{ \mathbf{X} \in \mathbb{E}^3 | |X_i| < 1 \} \]
be the reference configuration of a body. Consider the deformation
\[ \mathbf{x} = \boldsymbol{\phi}(\boldsymbol{\zeta}) = \frac{1}{6} \begin{pmatrix} -2 & -2 & 2 \\ 5 + 3 \, \sqrt{3} & 2 & -5 + 3 \, \sqrt{3} \\ 5 - 3 \, \sqrt{3} & 2 & -5 - 3 \, \sqrt{3} \end{pmatrix} \boldsymbol{\zeta} + \begin{pmatrix} 2 \\ 0 \\ 2 \end{pmatrix} \]
Tasks
- Compute and sketch the positions of the vertices \(a'\), \(b'\) and \(c'\) of the deformed configuration \(B' = \boldsymbol{\phi}(B)\) in the plot below
\[ c': \begin{pmatrix} 1.67 \\ 1.70 \\ 1.97 \end{pmatrix} \]
Compute the deformation gradient and the right Cauchy-Green tensor \(\mathbf{C} = \mathbf{F}^T \mathbf{F}\).
Compute the eigensystem of \(\mathbf{C}\) and use this to compute the right stretch tensor \(\mathbf{U} = \left(\mathbf{F}^T \mathbf{F}\right)^{\frac{1}{2}}\).
\[ \mathbf{U} = \frac{1}{6} \begin{pmatrix} 5 + 3 \, \sqrt(3) & 2 & -5 + 3 \, \sqrt(3) \\ 2 & 8 & -2 \\ -5 + \, \sqrt(3) & -2 & 5 + \, \sqrt(3) \end{pmatrix} \]
Compute the displacement gradient \(\mathbf{H}\). Afterwards, compute the constant and linear part of left Cauchy-Green tensor \(B\) and justify that the small-displacement limit.
Compute \(\mathbf{Q} = \mathbf{F} \mathbf{U}^{-1}\). Show that \(\mathbf{Q}\) is orthogonal.
\[ \mathbf{Q} = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix} \]
- Decompose \(\mathbf{F}\) into a rotation, isotropic dilation and volume preserving strain.
\[ det(F) = -3.464 \]